(1 (z)n+l - Project Euclid

ILLINOIS JOURNAL OF MATHEMATICS Volume 37, No. 3, Fall 1993

PLURIHARMONIC SYMBOLS OF COMMUTING TOEPLITZ OPERATORS Boo RIM CHOE AND YOUNG Joo LEE 1. Introduction and Results

Our setting throughout the paper is the unit ball Bn of the complex n-space cn; dimension n is fixed and thus we usually write B Bn unless otherwise specified. The Bergman space A2(B) is the closed subspace of L2(B) L2(B, V) consisting of holomorphic functions where V denotes the volume measure on B normalized to have total mass 1. For u L=(B), the Toeplitz operator Tu with symbol u is the bounded linear operator on A2(B) defined by Tu(f) P(uf)where P denotes the orthogonal projection of L2(B) onto A2(B). The projection P is the well-known Bergman projection which can be explicitly written as follows:

P()(z)

fB (1

(W)

(z)n+l dV(w)

(zB)

for functions 0 L2(B) Here (,) is the ordinary Hermitian inner product on C n. See [7, Chapters 3 and 7] for more information on the projection P. In one dimensional case, Axler and (ukovi6 [3] has recently obtained a complete description of harmonic symbols of commuting Toeplitz operators: if two Toeplitz operators with harmonic symbols commute, then either both symbols are holomorphic, or both symbols are antiholomorphic, or a nontrivial linear combination of symbols is constant (the converse is also true and trivial). Trying to generalize this characterization to the ball, one may naturally think of pluriharmonic symbols. A function u C2(B) is said to be pluriharmonic if its restriction to an arbitrary complex line that intersects the ball is harmonic as a function of single complex variable. As is well known, a real-valued function on B is pluriharmonic if and only if it is the real part of a holomorphic function on B. Hence every pluriharmonic function on B can be expressed, uniquely up to an additive constant, as the sum of a holomorphic function and an antiholomorphic function.

Received August 6, 1991. 1991 Mathematics Subject Classification. Primary 47B38; Secondary 32A37. (C)

1993 by the Board of Trustees of the University of Illinois Manufactured in the United States of America

424

425

COMMUTING TOEPLITZ OPERATORS

In the present paper we consider the same problem of characterizing pluriharmonic symbols of commuting Toeplitz operators on the ball. Our first result is a necessary condition in terms of ’-harmonicity (see Section 2 for relevant definitions) for such symbols.

,

THEOREM 1. Let f, g, h, and k be holomorphic functions on B such that and h + k are pluriharmonic symbols of two commuting Toeplitz operahy, is e/g-harmonic on B. tors on A2(B). Then

f+

The proof in [3] shows that the converse of Theorem 1 is also true in one dimensional case. Unfortunately, we were not able to prove or disprove the converse of Theorem 1 on the ball in general. However, Theorem 1 is enough to produce a simple characterization in case one of symbols is holomorphic (or antiholomorphic which amounts to considering adjoint operators). Its proof will make use of a recent characterization (see Proposition 7) of Ahern and Rudin [2] on /-harmonic products.

THEOREM 2. Suppose that u and v are pluriharmonic symbols of two commuting Toeplitz operators on A2(B). If u is nonconstant and holomorphic, then v must be holomorphic. Recall that a bounded linear operator on a Hilbert space is called normal if it commutes with its adjoint operator. Since the adjoint operator of the the Toeplitz operator with symbol u is the Toeplitz operator with symbol following is an immediate consequence of Theorem 2 whose proof is therefore omitted.

,

COROLLARY 3. The Toeplitz operator with holomorphic symbol u is normal on A2(B)

if and only if u is constant.

D

In Section 2 we collect some facts about /’-harmonic functions which are needed in Section 3 where we prove Theorems 1 and 2. In Section 4 we conclude the paper with some remarks and discussions related to the converse of Theorem 1 and a possible pluriharmonic version of Corollary 3. 2. /-Harmonic Functions

For z, w

B, z

z(W)

4=

z

0, define

-Izl-2(w, z)z

and q0(w) =-w. Then qz

V/1 -Izl (w -Izl-2(w, z)z) 2

1- (w,z)

’, the group of all automorphisms

426

BOO RIM CHOE AND YOUNG JOO LEE

’

(= biholomorphic self-maps) of B. Furthermore, each q has a unique representation q U rpz for some z B and unitary operator U on C". For u C2(B) and z B, we define

(au)(z)

z

where A denotes the ordinary Laplacian. The operator is called the invariant Laplacian because it commutes with automorphisms of B in the sense that (u q) (u)o q for q /. We say that a function u C2(B) is c/e-harmonic on B if it is annihilated on B by One can easily see that /-harmonic functions are precisely harmonic ones in one dimensional case. As is the case for harmonic functions, -harmonic functions are characterized by a certain mean value property (see [7, Chapter 4]): a function u C(B) is g-harmonic on B if and only if

z.

.

(uo r#)(O)

fs(UO p)(r) do’()

(0

_< r

< 1)

for every r# Here o- denotes the rotation invariant probability measure on the unit sphere S, the boundary of B. This is the so-called invariant mean value property. The following area version of this invariant mean value property also gives a characterization of ’-harmonicity of functions continuous up to the boundary (see [7, Proposition 13.4.4]): a function u C(B) is .e’-harmonic on B if and only if

(uo )(0)

fB(U

p) dV

for every p ’. The key step to our proof of Theorem 1 is adapted from that of [3]. That is, we will use a slight variant of the characterization of -harmonicity given by the area version of invariant mean value property. To state it, let us introduce some notations. We associate with each v C(B) its so-called radialization ,/(v) defined by the formula

’( v)( z)

fz( v

U)( z) dU (zeB)

where dU denotes the Haar measure on the group of all unitary operators on C n. Using Proposition 1.4.7 of [7], one can easily verify that

(v)(z)

fs(Izl) do’()

(z

B)

427


and hence W’(v) is indeed a radial function on B. We write ’(v) ’(v) has a continuous extension up to the boundary.

C( B )

PROPOSITION 4. Suppose that u on B if and only if

f,,(uo

(1)

dV

LI(B). Then u is

C(B) if

/g’-harmonic

(uo

and

(2)

’(uo p)

C(B)

for every Proof. We first prove the easy direction. Suppose that on B. Let q

u is ’-harmonic

’. By the invariant mean value property, we have

(uo q)(O)

fs(U

rp)(r() dr(()

for every r [0, 1). Integrating in polar coordinates, we have (1). The above also shows that z’(uo0) is constant on B, with value (u orp)(0), and therefore (2) holds. To prove the other direction (which we need for the proof of Theorem 1), suppose that (1) and (2) hold. Let q and put v ’(u q). We first show that v is ’-harmonic on B. Since v C(B) by (2), it is sufficient to show the area version of invariant mean value property of v. To do this, fix q ’. Then

fn( v

(3)

q) dV

fsf@(uo Fv) (z) dUdV(z)

q U 0 ’. For a fixed unitary operator U on C n, consider the inverse mapping Gv of Fv and put a Fv(0)= (q U q)(0). Then, since I-X(0)l-I(0)l, we have ([7, Theorem 2.2.5])

where Fv

"

1 -la[ 2

(1 -I o(0)12)(1 -I g,(O)12) (1 11 (-;--i;i-{ )(0))12

[q(0) [2)(1

428


On the other hand, we have [7, Theorem 2.2.6] n+l

JRGv( w)

.w, a-

,2

1 and suppose that the lemma is proved for rn 1. Let F (F1,..., Fm) and G (G1,..., Gm). We may assume that f contains the origin. We may further assume that IF(0)[ [G(0)[ 1. Pick unitary operators U and U2 on C m such that

UI(F(0))

U2(G(0))

(1, 0,..., 0). Let

UIF= (fl,’’’,fm) and U2oG (gl,’",gm)"

434

Then we have


E.m=f.3. Eim=lgigi on 12 and hence, by Lemma 9, E L.(z)f,.(w) E m

m

j=

j=l

for all z, w fl. Taking w 0, we obtain fl g on 12. Thus, by induction hypothesis, there exists some unitary operator U on C m-1 such that (f2,’", fro) U (g2,... gin) on f. Now let 1 0

0

0

v 0

Then U3 is a unitary operator on C m and we have F proof is complete. []

D

U1-1 U Uz G. The

In what follows, we let Vf= (Dxf,..., Dnf) and

denotes the differentiation with respect to tions, equation (11) becomes

(12)

f ET=lZiDif where z-variable. With these nota-

IVfl 2 + I,.gl 2 -IVgl 2 -4-I,.fl 2.

We assert the following: Suppose that (12) holds on B 2.

If Vf(0)

Vg(0) 4 0, then f

g on B 2.

Proof By (12) and Lemma 10 there is a unitary operator (aii) on C 3 such that

(13)

/

0/11

0/12

0/13

0/21 0/31

0/22 0/32

0/23 0/33

lg D2g

/

Vg(O) (1, 0)without loss of generality. Then, evaluating both sides of (13) at the origin, one can easily find that all 1 and 0. It follows that Dlf Dig. Hence Dzf- D2g 0/31 0/12 O/13 0/21 does not depend on Zl-variable. In order to prove D2f D2g it is therefore sufficient to show that Dzf(O, z 2) D2g(0, z 2) for Iz21 < 1. Evaluating both sides of (12) at points (0, z2), we obtain that .IDzf(0, z2)l ID2g(0, z2)l and thus there exists a unimodular constant h such that

Assume that Vf(O)

(14)

D2f(O, z2)

AD2g(O, z2)

435


for [Z21 < 1. Assume that both sides of (14) are not identically zero; otherwise we are done. By (13),

z2D2g(O, z2)

a32D2g(O z2) + a33z2D2f(O, z2).

Insert (14) into the above. A little manipulation yields a32 "-O23--’--0 and a33 a,. Thus, we have o@f )t.@g. Evaluating both sides of this at points (Zl, 0), we obtain A 1. The proof is complete. We now conclude the paper with another special

case"

If f + 0 and 4= 0 for some 1 < j < where fm denotes the ruth degree term in the homogeneous expansion of f on B, then (12) implies f h g for some unimodular constant h.

f.

Thus, if there were counter examples, then there would be no "gap" in their homogeneous expansions.

Proof

First note that the invariant mean value property of

Ill 2

yields lfml

do"

fsIgml

2

do"

(m

1,2,...)

where gm denotes the mth degree term in the homogeneous expansion of g on B. Hence gt+l 0 and gj 4= 0 by hypothesis. Now, by Lemma 10 as before, there exists a unitary operator U on C n/l such that (Vf,..@g)= U o(Vg, .@f). In particular, there are some vectors a,/3 C n and a constant h with la[ 2 / IAI 2 1/312 / IA[ 2 1 such that

,.@f= (Vg, a) + a,.g and ..g

(15)

(Vf,,8} + a._f.

If lal 1, then a =/3 0 and (15) shows o@f hg, hence f hg. So, we assume lal < 1 and derive a contradiction. Equate terms of the same degree in the homogeneous expansions of both sides of two equations of (15) to obtain

mfm

(Vgm+lm,ot) + ,mg m and mg m

(Vfm+l,/) + ,mfm,

so that

m(1 l*12)fm (Vgm+l, a) / ,k(Vfm+l

>

for m 1, 2, Since fl+ 0, the above shows that fm gl+ for all 1 < m < l, which is a contradiction. The proof is complete.

gm

0

436


We thank H. O. Kim for many helpful conversations about the material in Section 4. The first author was in part supported by the Korea Science and Engineering Foundation. REFERENCES

.

[1]. P. AHERN and K. JOHNSON, Differentiability criteria and harmonic functions on B n, Proc. Amer. Math. Soc., vol. 89 (1983), pp. 709-712. [2]. P. AHERN and W. RUDIN, e/g-harmonic products, Indag. Math., vol. 2 (1991), pp. 141-147. [3]. S. AXLER and UKOVI6, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory, vol. 14 (1991), pp. 1-11. [4]. R. GRAHAM, The Dirichlet problem for the Bergman Laplacian, Comm. Partial Differential Equations, vol. 8 (1983), pp. 433-476. [5]. W. RUDIN, Functional analysis, McGraw-Hill, New York, 1973. [6]. A smoothness condition that implies pluriharmonicity, unpublished. [7]. Function theory in the unit ball of C n, Springer-Verlag, New York, 1980. KOREA ADVANCED INSTITUTE SEOUL, KOREA

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